Physical fields can often be simulated over a user-defined domain—e.g., a surface or volume specified geometrically—by “discretizing” the domain (i.e., dividing it into discrete elements), and modeling the field-governing equations and applicable boundary conditions with a system matrix equation, i.e., a (typically large) linear system of equations that describe the behavior of the field within the discretized domain. For example, electromagnetic fields can be simulated using a discretized formulation of Maxwell's equations. To set up the matrix equation, the finite element method (FEM) is widely used because of its ability to model complex heterogeneous and anisotropic materials and to represent geometrically complicated domains using, for example, tetrahedral elements. FEM is a numerical technique for finding approximate solutions of partial differential equations (such as, e.g., Maxwell's equations), i.e., a technique that enables problems lacking exact mathematical (“analytical”) solutions to be approximated computationally (“numerically”).
In brief, FEM typically involves representing a surface or spatial volume as many small component elements. This discretization may be accomplished by defining a mesh grid (such as, e.g., a triangular, tetrahedral, or other polygonal mesh) over the domain. The physical fields (e.g., the components of electric and magnetic fields) may then be expressed in a form suitable to the discretized domain. For example, fields may be represented in a finite-dimensional function space of piecewise polynomial functions (e.g., piecewise linear functions) that can be described as linear combinations of basis functions, or “finite elements,” whose “support”—defined as the portion of the domain where the basis functions are non-zero—includes only a small number of adjacent meshes. The boundary value problem that describes the behavior of the fields in the domain (i.e., the field-governing equations and boundary conditions) is typically rephrased in its weak, or variational, form before discretization.
FEM results in a matrix equation which may then be solved with a direct or iterative solver, depending on the size and characteristics of the linear system. (A “solver,” as the term is used herein, denotes a method for solving a system of equations, or a computer program implementing such a method, as determined by context.) For large three-dimensional problems, direct solvers potentially require prohibitive amounts of memory and suffer poor parallel scalability. Therefore, iterative solvers typically present the only practical means for solving large systems. In iterative methods, the problem is approached in successive steps, with each step refining a previous approximation to more closely approach the exact solution. A preconditioner (i.e., a matrix that reduces the “condition number” of the problem, which, in turn, is a metric of the propagation of approximation errors during numerical solution) is often applied to the original system matrix to increase the rate of convergence toward the solution—i.e., to reduce the number of necessary iterations.
A powerful technique to facilitate parallel solution of large electromagnetic problems, which also provides an efficient and effective preconditioner, is the domain decomposition method (DDM). In this method, the original domain of the problem is decomposed into several (usually non-overlapping and possibly repetitive) subdomains; for example, a cuboid spatial domain may be divided into a series of smaller adjacent cubes. The continuity of electromagnetic fields at the interfaces between adjacent subdomains is enforced through suitable boundary conditions (also referred to as transmission conditions), which are preferably chosen so as to avoid mathematical complication (e.g., so that modeling of each subdomain involves a “well-posed” problem having an unambiguous solution, and such that convergence occurs rapidly enough to be computationally tractable).
For problems involving electromagnetic fields (or, more generally, vector wave problems), it has been shown that subdomain problems are well-posed and convergence occurs at an acceptable rate if so-called “Robin transmission conditions” are imposed on the boundaries between subdomains. The subdomain boundaries do not represent actual physical boundaries, and are introduced merely for computational convenience. Transmission conditions specify how the fields behave at those “virtual boundaries” to ensure that the solution obtained by domain decomposition is consistent with a solution for the undivided domain (i.e., a problem formulation wherein the boundaries do not exist). For example, transmission conditions may specify that the fields, or their derivatives, are continuous across the boundary. Robin transmission conditions express these requirements in terms of particular combinations of fields, currents, and their derivatives. First-order Robin transmission conditions have been successfully used in antenna-array-type problems, but have generally proven unsuitable for other problem categories because they damp only propagation errors, but not evanescent errors. Attempts to use higher-order Robin transmission conditions, on the other hand, have suffered from poor convergence properties, resulting from the high sensitivity of the convergence rate to the particular choice of certain coefficients contained in the conditions, hereinafter referred to as “Robin coefficients.”
Accordingly, there is a need for electromagnetic simulation methods that achieve high convergence rates for the simulated electromagnetic field across a broad range of applications. In particular, there is a need for DDMs that employ optimized Robin transmission conditions.